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Keywords = conditionally solvable model

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8 pages, 767 KB  
Communication
Exact Solutions, Critical Parameters and Accidental Degeneracy for the Hydrogen Atom in a Spherical Box
by Francisco M. Fernández
Physics 2025, 7(4), 48; https://doi.org/10.3390/physics7040048 - 15 Oct 2025
Viewed by 467
Abstract
This paper for the first time derives some properties of the hydrogen atom inside a box with an impenetrable wall. Scaling of the Hamiltonian operator proves to be practical for the derivation of some general properties of the eigenvalues. The radial part of [...] Read more.
This paper for the first time derives some properties of the hydrogen atom inside a box with an impenetrable wall. Scaling of the Hamiltonian operator proves to be practical for the derivation of some general properties of the eigenvalues. The radial part of the Schrödinger equation is conditionally solvable and the exact polynomial solutions provide helpful information. There are accidental degeneracies that take place at particular values of the box radius, some of which can be determined from the conditionally-solvable condition. Some of the roots stemming from the conditionally-solvable condition appear to converge towards the critical values of the model parameter. This analysis is facilitated by the Rayleigh–Ritz method that provides accurate eigenvalues. Full article
(This article belongs to the Section Quantum Mechanics and Quantum Systems)
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17 pages, 862 KB  
Article
Unified Algorithm of Factorization Method for Derivation of Exact Solutions from Schrödinger Equation with Potentials Constructed from a Set of Functions
by Raoul R. Nigmatullin and Airat A. Khamzin
Mathematics 2023, 11(18), 3822; https://doi.org/10.3390/math11183822 - 6 Sep 2023
Viewed by 1214
Abstract
We extend the scope of the unified factorization method to the solution of conditionally and unconditionally exactly solvable models of quantum mechanics, proposed in a previous paper [R.R. Nigmatullin, A.A. Khamzin, D. Baleanu, Results in Physics 41 (2022) 105945]. The possibilities of applying [...] Read more.
We extend the scope of the unified factorization method to the solution of conditionally and unconditionally exactly solvable models of quantum mechanics, proposed in a previous paper [R.R. Nigmatullin, A.A. Khamzin, D. Baleanu, Results in Physics 41 (2022) 105945]. The possibilities of applying the unified approach in the factorization method are demonstrated by calculating the energy spectrum of a potential constructed in the form of a second-order polynomial in many of the linearly independent functions. We analyze the solutions in detail when the potential is constructed from two linearly independent functions. We show that in the general case, such kinds of potentials are conditionally exactly solvable. To verify the novel approach, we consider several known potentials. We show that the shape of the energy spectrum is invariant to the number of functions from which the potential is formed and is determined by the type of differential equations that the potential-generating functions obey. Full article
(This article belongs to the Special Issue Applications of Mathematical Methods in Quantum Mechanics)
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